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Hard sphere fluids in annular wedges: density distributions and
depletion potentials
V. Botan, F. Pesth, T. Schilling, and M. Oettel∗
Johannes–Gutenberg–Universitat Mainz,
Institut fur Physik, WA 331, D–55099 Mainz, Germany,
Abstract
We analyze the density distribution and the adsorption of solvent hard spheres in an annular
slit formed by two large solute spheres or a large solute and a wall at close distances by means
of fundamental measure density functional theory, anisotropic integral equations and simulations.
We find that the main features of the density distribution in the slit are described by an effective,
two–dimensional system of disks in the vicinity of a central obstacle. This has an immediate
consequence for the depletion force between the solutes (or the wall and the solute), since the
latter receives a strong line–tension contribution due to the adsorption of the effective disks at the
circumference of the central obstacle. For large solute–solvent size ratios, the resulting depletion
force has a straightforward geometrical interpretation which gives a precise “colloidal” limit for
the depletion interaction. For intermediate size ratios 5 . . . 10 and high solvent packing fractions
larger than 0.4, the explicit density functional results show a deep attractive well for the depletion
potential at solute contact, possibly indicating demixing in a binary mixture at low solute and high
solvent packing fraction.
∗Electronic address: oettelm@uni-mainz.de
1
I. INTRODUCTION
The equilibrium statistical theory of inhomogeneous fluids whose foundations were laid
out in the 1960’s [1, 2] is a well–studied subject which has been developed since by a fruitful
interplay between simulations and theory. For the simplest molecular model, the hard–body
fluid mixture, this has led to the development of powerful, geometry–based density function-
als [3, 4, 5, 6, 7, 8] which accurately describe adsorption phenomena, phase transitions (such
as freezing for pure hard spheres or demixing for entropic colloid–polymer mixtures) and
molecular layering near obstacles. Such a class of density functionals has not been found yet
for fluids with attractions, however, significant progess has been achieved in the description
of bulk correlation functions and phase diagrams through the method of integral equations
[9, 10, 11].
Strong inhomogeneities occur if fluids are confined on molecular scales, a topic which
enjoys continuous interest [12, 13]. For example, the packing of molecules between parallel
walls leads to oscillating forces between them which can be measured experimentally and
determined theoretically [14]. With the development of preparational techniques for colloids
and their mixtures, the “molecular” length scale has been lifted to the range of several nm
up to µm which even allows the direct observation of modulated density profiles through
microscopy besides the measurement of resulting forces on the walls. Furthermore, in the
colloidal domain the possibility to tailor the interparticle interactions to a certain degree
allows the close realization of some of the favourite models for simple fluids fancied by theo-
rists, such as e.g. hard spheres [15]. This has opened the route to quantitative comparisons
between experiment, theory and simulations.
In an asymmetric colloidal mixture with small “solvent” particles and at least one species
of larger solute particles, the phenomenon of solvent–mediated, effective interactions between
the solute particles may give rise to various phase separation phenomena [16]. From a
theoretical point of view, these effective interactions are interesting since they facilitate
the description of mixtures in terms of an equivalent theory for one species interacting by
an effective potential [17, 18]. If the solvent particles possess hard (or at least steeply
repulsive) cores, they are excluded from the region between two solute particles if the latter
are separated by less than one solvent diameter. This gives rise to strong depletion forces
whose understanding is crucial for the concept of an effective theory containing only solute
2
mid−plane between colloids
zone
overlap
quasi−twodimensional gas of hard disks
solute surface
edge of wedge
FIG. 1: (color online) The annular wedge which is formed between two large solute particles for
separations h ≤ σ. Black areas denote domains which are forbidden for the centers of the solvent
spheres.
degrees of freedom. Since the magnitude of the depletion force is directly linked to the
solvent density distribution around the solutes, the quantitative investigation of the solvent
confined between solute particles appears to be important. The confinement becomes rather
extreme for large asymmetry between solute and solvent (see below).
In the following we want to concentrate on the idealized system of additive hard spheres
and in particular on the effective interaction between two solute particles in solvent, i.e. the
case of infinite dilution of solute particles in the colloidal mixture. The case of the interac-
tion of one solute particle with a wall is a special case in that the radius of the other solute
particle goes to infinity. For the case of hard spheres (solvent diameter σ = 2Rs, solute ra-
dius Rb such that the size ratio is α = Rb/Rs and R = Rb +Rs is the radius of the exclusion
sphere for solvent centres around a solute sphere) it has been found that the theoretical
description using “bulk” methods becomes increasingly inaccurate for size ratios α & 5 and
solvent densities ρ∗s = ρsσ
3 & 0.6, when compared to simulations. Here, the terminus “bulk”
methods refers to methods which determine the bulk pair correlation function between the
colloids, gbb(r), from which the depletion potential is obtained as βW = − ln gbb. Bulk inte-
3
gral equation (IE) methods [19] and the “insertion” trick in density functional theory (DFT)
[20, 21, 22, 23] fall into this category. For larger size ratio, one would expect that the well–
known Derjaguin approximation [24, 25, 26] becomes accurate very quickly. Interestingly,
however, both the mentioned bulk methods and simulations deviate significantly from the
Derjaguin approximation (besides disagreeing with each other) for size ratios 10 and solvent
densities ρ∗s = 0.6 . . . 0.7 [26, 27, 28] when the surface–to–surface separation h between the
solute particles is close to σ (h . σ, i.e. at the onset of the depletion region where the solvent
particles are “squeezed out” between the solute particles). One notices that for h ≤ σ the
solute particles form an annular wedge with a sharp edge which restricts the solvent particles
to quasi–2d motion (see Fig. 1). Taking this observation into account, the phenomenological
analysis of Ref. [26] predicted that the solvent adsorption at this edge leads to a line contri-
bution to the depletion potential which is proportional to the circumference of the circular
egde and thus to R1/2. Such a term in the depletion potential causes a very slow approach
to the Derjaguin limit for large solutes (the Derjaguin approximation describes the deple-
tion potential essentially by volume and area terms of the overlap of the exclusion spheres
pertaining to the solutes [25, 26], see below). In this way, a generalized Derjaguin approx-
imation serving as a new “colloidal” limit can be formulated which turns out to have a far
more general meaning [29] than anticipated. Using the concept of morphological (morpho-
metric) (morphometric) (morphometric) (morphometric) (morphometric) (morphometric)
(morphometric) (morphometric) thermodynamics introduced in Ref. [30], one finds that the
insertion free energy of two solute particles (and thus the depletion potential between them)
only depends on the volume, surface area, and the integrated mean and Gaussian curvatures
of the solvent accessible surface around the two solutes. The coefficients of these four terms
depend on the solvent density but not on the specific type of surface. In this manner, the
phenomenological line tension of Ref. [26] is related to the general coefficient of mean cur-
vature of the hard–sphere fluid [29], and the Derjaguin approximation is equivalent to the
morphometric analysis restricted to volume and surface area terms.
Since the depletion force in the hard–sphere system is directly linked to an integral over
the solvent contact density on one solute (see Eqs. (1) and (2) below), one should be able to
connect the morphometric approach to features of the solvent density profile in the annular
wedge. This is a strong motivation for us to explicitly determine the wedge density profile.
We will do so by means of density functional theory and anisotropic integral equations, and
4
compare it to results of Monte–Carlo (MC) simulations for selected parameters. For the
intermediate densities ρ∗s = 0.6 and 0.7 and various size ratios between solute and solvent
results for the depletion force from such explicit DFT calculations were already presented
in Ref. [29], and good agreement with the morphometric depletion force was obtained for
size ratios between 5 and 40. Here, we will present a detailed analysis of the wedge density
profiles and consider also higher densities. We will give a thorough comparison to recent
results from MC simulations which have been obtained for the wall–solute interaction at
a solvent density ρ∗s = 0.764 (ηs = (π/6)ρsσ
3 = 0.4) and size ratios between 10 and 100
[31, 32, 33].
The paper is structured as follows. In Sec. II we introduce the basic notions of density
functional and integral equation theory and present a short description of the Monte Carlo
simulations employed here. In Sec. IIIA we analyze in detail the density profiles and the
corresponding depletion forces for the wall–sphere geometry for the particular solvent density
ρ∗s = 0.764. This geometry permits explicit calculations up to solute–solvent size ratios 100
and a test of the “colloidal limit” of the morphometric depletion force. Sec. III B gives an
analysis of the depletion force and potential in the sphere–sphere geometry (corresponding
to the important case of the effective interaction in a dilute mixture) for intermediate size
ratios 5 and 10 and higher solvent densities ρ∗s = 0.8 . . . 0.9.
II. THEORY AND METHODS
We consider two scenarios (see Fig. 2): (a) One hard solute sphere immersed in the hard
sphere solvent of density ρs confined to a slit, created by two hard walls at distance L.
The surface–to–surface distance between one wall and the solute sphere is denoted by h,
furthermore L ≫ h such that the correlations from one wall do not influence the solute
interaction with the other wall. (b) Two hard solute spheres immersed in the bulk solvent
spheres with surface–to–surface distance h. The solvent accessible surface is given by the
dashed lines in Fig. 2, thereby one sees that for h ≤ σ the two solutes (or the solute and
one wall) form an annular wedge in which the solvent adsorbs. The solvent density profile
ρ(r) ≡ ρ(r||, z) depends only on z, the coordinate on the symmetry axis and the distance r||
to the symmetry axis. In the latter case (b), when the first colloid is centered at the origin
and the second one at z = 2Rb + h, the total force f(h) on the first colloid is obtained by
5
integrating the force −∇ubs(r) (ubs is the solute–solvent potential) between the solute and
one solvent sphere over the density distribution ρ(r):
βf(h) = −∫
d3r ρ(r) ez · ∇ubs(r) (1)
= 2πR2
∫ 1
−1
d(cos θ) cos θ ρ(r) ,
[|r| = R , cos θ = r · ez] .
Thus the total depletion force reduces to an integral over essentially the contact density on
the exclusion sphere around the colloid. This follows from exp(−βubs)β∇ubs = −rδ(|r|−R)
and the observation that ρ(r) exp(βubs) is continuous across the exclusion sphere surface. In
case (a), the force on the colloid is the negative of the excess force fw on the wall and the
excess force is determined by the total force on the wall minus the contribution from the
bulk solvent pressure p. Through the wall theorem, the latter is given by βp = ρw where ρw
is the contact density of the solvent at a single wall. By an argument similar to the above
one and putting the exclusion surface of the wall at z = 0, fw is determined as
βfw(h) = −2π
∫ ∞
0
r|| dr|| (ρ(r||, z = 0) − ρw) . (2)
Thus, fw is equivalent to the excess adsorption at the wall which makes it somewhat easier
to determine in MC simulations than f [32].
A. Density functional theory
The equilibrium solvent density profile ρ(r) ≡ ρeq(r) can be determined directly from the
basic equations of density functional theory. The grand potential functional is given by
Ω[ρ] = F id[ρ] + F ex[ρ] −∫
dr(µ − V ext(r)) , (3)
where F id and F ex denote the ideal and excess free energy functionals of the solvent. The
solvent chemical potential is denoted by µ and the solute(s) and/or the walls define the
external potential V ext. The ideal part of the free energy is given by
βF id =
∫
dr ρ(r)(
ln(ρ(r)Λ3) − 1)
, (4)
with Λ denoting the de–Broglie wavelength. The equilibrium density profile ρeq(r) for the
solvent at chemical potential µ = β−1 ln(ρs Λ3) + µex (corresponding to the bulk density ρs)
6
(a)
(b)
FIG. 2: View of the geometric configurations used in this work. (a) Solute sphere of radius Rb
immersed in a solvent–filled slit of width L. Only the density profile in the annular wedge between
the left wall and the solute is of interest since it determines the depletion force between solute and
one wall. Note that one can also determine the slit density profile and the corresponding depletion
force for 2Rb > L (i.e. when the solute does not fit into the slit) as long as L is large enough that
the correlations from the right wall do not reach into the annular wedge. (b) Two solute spheres of
radius Rb at distance h immersed in bulk solvent. For both setups, the solvent sphere diameter is
given by σ, and the radius of the exclusion sphere around a solute particle is given by R = Rb+σ/2.
is found by minimizing the grand potential in Eq. (3):
lnρeq(r)
ρs+ βV ext(r) = −β
δF ex[ρeq]
δρ(r)+ βµex . (5)
For an explicit solution, it is necessary to specify the excess part of the free energy. Here
we employ two functionals of fundamental measure type (FMT). These are the original
Rosenfeld functional (FMT–RF) [3] and the White Bear functional (FMT–WB, with Mark
I from Refs. [4, 5] and Mark II from Ref. [6]), for another closely related variant see Ref. [7].
It has been demonstrated that FMT gives very precise density profiles also for high densities
of the hard sphere fluid in various circumstances. Furthermore, the White Bear II functional
7
possesses a high degree of self–consistency with regard to scaled–particle considerations [6].
However, we do not consider the tensor–weight modifications of these functionals which
are necessary to obtain a correct description of the liquid–solid transition [34] and are of
higher consistency in confining situations which reduce the dimensionality of the system
(“dimensional crossover”, see Ref. [35, 36]). This might be an issue in some circumstances
(see below).
Th explicit forms of the excess free energy are given in App. B. Numerically, due to the
non–local nature of F ex, Eq. (5) corresponds to an integral equation for the density profile
depending on the two variables r|| and z. The sharpness of the annular wedge in the solute–
solute and the solute–wall problems for high size ratios necessitates rather fine gridding
(which makes the calculation of F ex and δF ex/δρ very time–consuming) and introduces an
unusual slowing–down of standard iteration procedures of Picard type. To overcome these
difficulties, we made use of fast Hankel transform techniques and more efficient iteration
procedures. These are described in detail in App. B as well.
We remark that there have been earlier attempts to obtain explicit density distributions
using DFT around two fixed solutes and thereby to extract depletion forces [37, 38, 39]. In
Ref. [38] this was done using the simple Tarazona I functional for hard spheres [40], and
considerations were limited to a small solvent packing fraction of 0.1 and solute–solvent size
ratios of 5. In Ref. [39], a minimization of FMT–RF was carried out using a real–space
technique for solvent densities up to ρ∗s = 0.6 and size ratios 5. In that respect, the present
technique is superior in that it allows us to present solutions for solvent densities up to
ρ∗s = 0.9 and size ratios up to 100.
B. Integral equations
The central objects within the theory of integral equations are the correlation functions
on the one and two–particle level in the solvent. We are interested in the two–particle
correlation functions in the presence of an external background potential (one fixed object,
solute or wall), from which the explicit solvent density profiles around the two fixed objects
(solute–solute or solute–wall) follow (see below). This is usually referred to as the method of
inhomogeneous or anisotropic integral equations, first employed for Lennard–Jones fluids on
solid substrates [41, 42] and for hard–spheres (HS) fluids in contact with a single hard wall
8
in Ref. [43]. Liquids confined between two parallel walls (a planar slit) have been extensively
studied by Kjellander and Sarman [44, 45, 46].
For an arbitrary background potential V (r) which corresponds to an equilibrium back-
ground density profile for the solvent ρV (r), the pair correlation function gij(r, r0) =
hij(r, r0) + 1 describes the normalized probability to find a particle of species i at posi-
tion r if another particle of species j is fixed at position r0. For our system, the species
index is either s (solvent particle) or b (big solute particle). In the solute–solute case, V is
given by the potential of one solute particle, whereas in the solute–wall case it is the poten-
tial of the wall. Then the equilibrium density profile discussed in the previous subsection
is related to the pair correlation function through ρ(r) = ρV (r) gsb(r, r0) (r0 specifies the
position of the (other) solute particle). The depletion force follows then through Eqs. (1)
and (2).
The corresponding direct correlation functions of second order cij(r, r0) are related to hij
through the inhomogeneous Ornstein–Zernike (OZ) equations
hij(r, r0) − cij(r, r0) =∑
k=b,s
∫
dr′ρV,k(r′)hik(r, r
′)ckj(r′, r0) (6)
In the dilute limit for the solute particles which we consider here, the background density
ρV,b for the solutes is zero, therefore the OZ equations reduce to
hss(r, r0) − css(r, r0) =
∫
dr′ρV (r′)hss(r, r′)css(r
′, r0) , (7)
hbs(r, r0) − cbs(r, r0) =
∫
dr′ρV (r′)hbs(r, r′)css(r
′, r0) . (8)
The background density profile is linked to css through the Lovett–Mou–Buff–Wertheim
equation [47]:
∇ρV (r) = −βρV (r)∇V (r) + ρV (r)
∫
dr′css(r, r′)∇ρV (r′) . (9)
A third set of equations is necessary to close the system of equations. The diagrammatic
analysis of Ref. [1] provides the general form of this closure which reads:
ln gij(r, r0) + βuij(r− r0) = hij(r, r0) − cij(r, r0) − bij(r, r0) , (10)
where uij are the pair potentials in the solute–solvent mixture and bij denote the bridge
functions specified by a certain class of diagrams which, however, can not be resummed in a
9
closed form. For practical applications, these bridge functions need to be specified in terms
of hij and cij to arrive at a closed system of equations. Among the variety of empirical forms
devised for this connection we mention those which, in our opinion, rely on somewhat more
general arguments. These are the venerable hypernetted chain (HNC), Percus–Yevick (PY)
closure and the mean spherical approximation (MSA) which can be derived by systematic
diagrammatic arguments [48], and the reference HNC closure which introduces a suitable
reference system for obtaining bij with subsequent free energy minimization [49]. (For bulk
properties of liquids, the self–consistent Ornstein–Zernike approximation (SCOZA) [9] and
the hierarchical reference theory (HRT) [10] are very successful. They rely on a closure of
MSA type but it appears to be difficult in generalizing them to inhomogeneous situations
such as considered here.) In calculations, we considered the closures
bij = γij − ln(1 + γij) (PY) , (11)
bij = γij − ln
[
1 +exp[1 − exp(−ξijr)]γij − 1
1 − exp(−ξijr)
]
(RY) , (12)
bij =1
2
γ2ij
1 + αijγij
(MV) . (13)
where γij = hij − cij . The Percus–Yevick (PY) approximation Eq. (11) is exactly solvable in
the bulk case even for the muticomponent HS fluid [50]. However, beyond providing good
qualitative behavior PY does not produce precise quantitative results in general, since it fails
at contact, where the value of pair distribution function is too small. For bulk systems, PY
generates a noticeable thermodynamic inconsistency, i.e. a discrepancy between different
routes to the equation of state. To overcome these deficiencies a refined approximation to
the closure, which interpolates between HNC (bij = 0) and PY closures was suggested by
Rogers and Young (RY) [51], see Eq. (12) where the ξij are adjustable parameters, which
can be found from the thermodynamic consistency requirement [51]. An even more suitable
for the asymmetric HS mixtures variant of the modified Verlet (MV) closure was suggested
in Ref. [52], see Eq. (13). Here the parameters αij are chosen to satisfy the exact relation
between bij(0) and the third virial coefficient at low densities. For the choice of the auxiliary
parameters ξij and αij in the present work, see App. C
We have solved the set of equations (7)–(13) for the solute–wall case where we could
employ similar numerical methods as in the numerical treatment of density functional theory.
(This is not possible for the solute-solute case, for an efficient method applicable in this case,
10
see Ref. [53].) In the solute–wall case, the background density profile ρV (r) ≡ ρV (z) depends
only on the distance z from the wall. The two–particle correlation functions depend on the
z–coordinates of the two particles individually and the difference in the radial coordinates:
gij(r, r0) ≡ gij(z, z0, r|| − r||,0). Thus, the Ornstein–Zernike equations (7) and (8) become
matrix equations for the correlation functions in the z–coordinates, and are diagonal for the
Hankel transforms of the correlation functions in the r||–coordinates. For more details on
the numerical procedure, see App. C.
C. Simulation Details
The Monte Carlo simulations for the wall–solute system were performed at fixed particle
number and volume of the simulation box. The upper boundary of the box was given by a
large hard sphere, the lower boundary by a planar hard wall (see hatched regions in Fig. 3).
All remaining boundaries were treated as periodic. We sampled the configuration space of
the small spheres by standard single particle translational moves. In order to impose the
asymptotic solvent bulk density ρ∗s = 0.764 (ηs = 0.4), the concentration of small spheres
in the box was set such that the density at contact with the hard wall far away from the
wedge (i. e. in the region marked by grey squares in Fig. 3, averaged over a depth 0.02
σ) settled to ρWall = 4.88 within 1% error. This value ρWall = 4.88 was obtained from the
density profile of hard spheres at a hard wall at the bulk packing fraction 0.4, calculated with
FMT–WBII which is very accurate. System sizes ranged from 1800 particles for Rb = 10 σ
to 8000 particles for Rb = 25 σ. In order to access the configurations inside the narrow part
of the wedge with sufficient accuracy, large numbers of Monte Carlo sweeps were required.
We equilibrated the systems for 5 × 105 MC sweeps (i. e. attempted moves per particle)
each. For data acquisition, we performed between 107 and 108 sweeps (depending on the
parameters h and Rb) and averaged over 106 to 107 samples. Note that from the simulations
we obtained only the wedge density profiles (see Figs. 5–7 below) but did not attempt to
obtain the depletion force from Eq. (2) where one needs an excess adsorption integral at the
wall. This would require considerably more sweeps [32, 33], see also Fig. 8 below for the
statistical errors of the simulated depletion force according to Ref. [32].
11
FIG. 3: Sketch of the simulation box. Hard walls are marked by hatches. The remaining boundaries
were treated as periodic. The grey squares mark the regions in which the density at wall contact
was sampled.
III. RESULTS
A. Wall–solute interaction
In this section, we analyze case (a), the problem of a solute sphere close to a wall. Previous
simulation work [31, 32, 33] has concentrated on the particular state point ρ∗s = 0.764
(packing fraction ηs = 0.4), therefore we present explicit results for the density profiles also
for this state point to facilitate comparison.
1. Density profiles
In Fig. 4, we show DFT results for the solvent density profile between wall and solute for
a solute–solvent size ratio of 20 and solvent–wall distance h = 0 (i.e. contact between wall
and solute molecular surface; left panel) and h = σ (i.e. the end of the depletion region;
right panel). There is strong adsorption at the apex of the annular wedge, given by the
coordinates z = 0 and r|| = r0 =√
2R(h − σ) − (h − σ)2, as reflected by the main peak.
Along the wall (z = 0), we observe strong structuring which is mainly dictated by packing
considerations. The apex peak corresponds to a ring of solvent spheres centered at r|| = r0
(for h = 0) or just one sphere near r|| = 0 (for h = σ). The second–highest peak in both
12
FIG. 4: (color online) Density profiles from FMT–WBII of solvent spheres confined to the annular
wedge between a solute sphere (size ratio Rb/Rs = 20) and a wall. The bulk solvent density is
ρ∗s = 0.764, and the wall separation L = 26σ (see Fig. 2 (a)). Grid resolution was equidistant in z
with ∆z = 0.002 σ and equidistant in x = ln(r||/σ) with ∆x = 0.005 (see App. B 2). Left panel:
solute–wall distance h = 0, right panel: h = σ.
panels of Fig. 4 appears where the distance between wall and solute sphere, measured along
the z–axis, is approximately 2σ and thus two rings of spheres fit between solute and wall.
However, for h = 0 (left panel of Fig. 4) packing in r||–direction leads to further structuring
of the density profile close to the apex of the wedge.
The integral of the solvent density at the wall (i.e. the adsorption at the wall) directly
determines the depletion force, see Eq. (2). Therefore, we analyze the density at the wall
further. In Fig. 5 we show ρ(r||, z = 0) computed with FMT–RF, FMT–WBII, IE–MV and
compare the results to the simulation data of Ref. [32]. Overall, DFT performs quite well but
it tends to overestimate the structuring of the profile. The currently most consistent version,
FMT–WBII, improves over FMT–RF particularly in this respect. The integral equation
results for the different closure are very similar to each other and are approximately of the
same quality as the solutions of FMT–RF. Note that the simulation data of Ref. [32] have
been averaged over a distance of 0.02 σ from the wall. This average was also applied to the
DFT data (which are computed with mesh size ∆z = 0.002 σ), whereas for the IE results we
13
4 5 6 7 8 9 10r|| / σ
1
10
100
ρ∗WBIIRFMC [32]IE-MV
0 1 2 3 4 5 6 7 8r|| / σ
1
10
100
ρ∗
WBIIRFMC [32]MCIE-MV
FIG. 5: (color online) Comparison between the various theoretical methods and MC simulations
(ours and from Ref. [32]) for the density at the wall (averaged over a distance of 0.02 σ). Left
panel: solute–wall distance h = 0, right panel: h = σ. The solute–solvent ratio is 20 and the
solvent bulk density is ρ∗s = 0.764. Note the logarithmic scale on the density axis.
show a suitable weighted average of the zeroth and first bin (∆z = 0.05), assuming linear
interpolation. As seen in the right panel of Fig. 5, IE–MV gives a much lower density close
to the apex of the wedge compared to the other methods. This is presumably due to the
lower resolution in z–direction which is possible for IE (see App. C).
To further investigate the issue of the dominant packing mode in the annular wedge, we
performed DFT calculations also for the additonal size ratios 10, 30, 50 and 100 for the
same solvent density ρ∗s = 0.764. Packing in r||–direction can be monitored conveniently
by viewing the annular wedge close to the apex as a quasi–2d system (see Fig. 1 and the
remarks in the Introduction). We define an effective, two–dimensional solvent density by
ρ2d(r||) =
∫ zs(r||)
0
dz ρ(r||, z) , (14)
where zs(r||) = R − (σ − h) −√
R2 − r2|| ≈ r2
||/(2R) − (σ − h) defines the surface of the
exclusion sphere around the solute (and thus the radius–dependent width of the annular
wedge). In Fig. 6 we show results for ρ2d(r||) for two configurations with slightly different
solute–wall distances: h = 0.95 σ (left panel) and h = σ (right panel). In the latter case,
one solvent sphere fits exactly between solute and wall at the apex, nevertheless ρ2d vanishes
there due to the vanishing slit width, zs(r|| → 0) → 0. Note that this is also a consequence
of an application of the potential distribution theorem [54] which states that the 3d density
14
reaches a finite, maximum value of exp(βµex) in a planar slit of vanishing width which is
equivalent to a vanishing 2d density (µex is the excess chemical potential of the bulk solvent
coupled to the slit). However, when the slit widens, the 2d density quickly rises [26]. For
h = σ (right panel in Fig. 6), this leads to the picture of an effective 2d system with a
small, soft and repulsive obstacle centered at r|| = 0 which induces moderate layering in the
2d–“bulk” with an effective “bulk” density ρ2d,s ≈ 0.7 . . . 0.8. For h = 0.95 σ (left panel in
Fig. 6), the obstacle in the center has a radial extent of r0 ≈√
Rσ/10 which ranges from
0.75 σ (size ratio 10) to 2.25 σ (size ratio 100) and induces stronger layering in the effective
2d system. The oscillations occur around the same “bulk” density as in the case h = σ.
The simulation results show weaker oscillations than the DFT results. Since the first peak,
e. g., occurs at wedge widths < 0.03 σ, we are testing the dimensional crossover properties of
DFT–FMT from 3d to 2d with a sensitive probe. The difference between the results of the
simulations and the DFT versions employed here are in line with previous investigations on
the dimensional crossover properties of FMT [36]. There it has been found that the strict
2d limit of FMT–RF results in a somewhat peculiar (integrable) divergence in the hard disk
direct correlation function c2d(r) for r → 0 and overestimated peaks in the corresponding
structure factor. The tensor weight modifications introduced in Ref. [34] result in better
dimensional crossover properties [56] and might improve the DFT results close to the apex
of the annular wedge. A more detailed investigation of tensor–weighted FMT with regard
to the correlations in narrow planar slits and also in the annular wedges considered here is
certainly of interest (although the algebra of App. B becomes much more extended in this
case).
Our observations should be compared with the phenomenological modelling of Ref. [26]
(“annular slit approximation”). There, the effective 2d system was approximated by an
idealized system of hard disks, layering around a hard cavity of radius r0 centered around
r|| = 0. The bulk density ρ2d,s of this hard disk system was determined via the self–
consistency condition p2d(ρ2d,s) = −2γ(ρs) where p2d is the 2d pressure and γ is the surface
tension of a hard wall immersed in a bulk solvent of density ρs. Employing scaled particle
theory [55], this yields ρ∗2d,s ≈ 0.66 for ρ∗
s = 0.764. This value is a bit smaller than the one
inferred from the 2d density profiles of Fig. 6. The main difference between the idealized
hard disk system of Ref. [26] and the effective 2d system as reflected in Fig. 6 lies probably
in the softness of the central cavity obstacle and also the softness of the effective particle
15
0 1 2 3 4 5 6(r
||-r
0) / σ
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ρ∗ 2d
size ratio 20 30 50
0 1 2 3 4 5 6 7r|| / σ
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ρ∗ 2d
size ratio 20 30 50
FIG. 6: (color online) Effective two–dimensional solvent density in the annular wedge for various
solute–solvent size ratios. Lines correspond to DFT results (FMT–WBII) and symbols show results
of our MC calculations. Left panel: solute–wall distance h = 0.95σ, right panel: h = σ. The solvent
bulk density is ρ∗s = 0.764. The curves are plotted up to the point where the width of the annular
wedge reaches the value of σ.
interactions due to the widening of the slit. Apart from that the overall picture of Ref. [26]
is confirmed well. In particular, this implies an important observable consequence: The
cavity circumference (of approximate length 2πr0) induces a line contribution 2πr0 γ2d(ρ2d,s)
to the insertion free energy of the solute near the wall. Since the insertion free energy is
equivalent to the depletion potential up to an additive constant, the latter acquires a term
∝√
R(σ − h), the corresponding term in the depletion force is ∝√
R/(σ − h). (Note that
for h → σ the cavity radius should stay finite as reflected in Fig. 6 (right panel), thus the
divergence of the depletion force there according to the geometrical argument of equating
the cavity radius with r0 is unphysical.) The interpretation of this line energy term within
the more general framework of morphometric thermodynamics will be given below.
We have seen that in radial direction, the packing of the solvent spheres close to the
apex of the annular wedge is well understood by the quasi–2d picture developed above. For
the particular point h = σ, there is only moderate layering in r||–direction, thus the full
density profile should be determined mainly by packing in z–direction. Indeed, for h = σ
we observe a quite remarkable collapse of the density profiles at the wall when plotted as
a function of r = r||/√
Rσ, see Fig. 7, indicating the relative unimportance of packing in
16
0 1 2 3
x = r|| / (Rσ)1/2
2
4
6
8
10
12
14
ρ∗
size ratio 20 50 100
0 0.1 0.2 0.3 0.4x
0
50
100
150
200
250
ρ∗
size ratio 20 50 100
0 20 40 60 80 1002R / σ
0
1
2
3
4
5
6
7
8
9
10
βσ2
f w /
(2R
)
WBIIMCDerjaguinupper bound annular slit approx.
FIG. 7: (color online) Left panel: density at the wall for solute–wall distance h = σ and the three
solute–solvent size ratios 20, 30 and 100. Lines are FMT–WBII data, symbols are MC simulation
results from Ref. [33]. Right panel: The depletion force between solute and wall at h = σ. The
solvent bulk density is ρ∗s = 0.764.
radial direction. According to Eq. (2), perfect scaling would imply that the depletion force
at h = σ is proportional to R. The DFT data violate scaling for r < 0.2 (i.e. when the slit
width is smaller than 0.02 σ), as can be seen from Fig. 7 (left panel, inset). This leads to
a slow increase of the scaled depletion force fw/(2R), see Fig. 7 (right panel). In contrast,
the simulation results of Ref. [33] indicate that fw/(2R) stays constant for large R, quite in
agreement with an upper bound derived within the annular slit approximation of Ref. [26]
which reads fw/(2R) = −2πγ(ρs)(1−η2d) where η2d = (π/4) ρ2d,sσ2 is the 2d “bulk” packing
fraction. Note that the value of the Derjaguin approximation at h = σ, fw/(2R) = −2πγ(ρs)
is completely off the simulation as well as the DFT values (see next subsection).
2. Geometric interpretation of the depletion force
Morphological (morphometric) thermodynamics [30] provides a powerful interpretation
of the depletion interaction, as has been shown very recently in the special case of the solute–
solute interaction [29]. Up to a constant, the depletion potential is nothing but the solvation
free energy of the wall and the solute. In morphological thermodynamics the solvation free
energy Fsol of a body is separated into geometric measures defined by its surface. As the
17
surface, we take the solvent–accessible surface, i.e. the body surface of the combined wall–
solute object is defined by the dashed lines in Fig. 2 (a). These geometric measures are
the enclosed volume V , the surface area A, the integrated mean and Gaussian curvatures C
and X, respectively. To each measure there is an associated thermodynamic coefficient: the
pressure p, the planar wall surface tension γ and two bending rigidities κ and κ such that
Fsol = pV + γA + κC + κX. (15)
Due to the separation of the solvation free energy into geometrical measures and geometry
independent thermodynamic coefficients, it is possible to obtain the coefficients p, γ, κ and
κ in simple geometries. The pressure p is a bulk quantity of the fluid. The surface tension γ
accounts for the free energy cost of forming an inhomogeneous density distribution close to
a planar wall. If the wall is curved the additional free energy cost is measured by κ and κ.
It is possible to obtain all four coefficients from a set of solvation free energies of a spherical
particle with varying radius. For a hard-sphere solute in a hard-sphere solvent very accurate
analytic expressions for the thermodynamic coefficients are known from FMT–WBII [6], see
also App. A.
For the particular case considered here, the depletion potential W (h) follows by subtract-
ing the sum of the solvation free energies of a single wall and a single solute sphere. Thus
we find
W (h < σ) = −p∆V − γ∆A − κ∆C − 4πκ (16)
where ∆V and ∆A are the volume and surface area of the overlap of the exclusion (depletion)
zones around wall and solute, respectively, and ∆C is the integrated mean curvature of that
overlap volume. Note that the fourth characteristic, the integrated Gaussian curvature is
the Euler characteristic which is 4π for wall and solute at h ≤ σ (one connected body) and
8π for wall and solute at h > σ (two disconnected bodies). This explains the last term in
18
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6h / σ
-10
-5
0
5
10
βσ2
f w /
(2R
)MCWBIImorphDerjaguinWBII (ins)IE-MV
x
0 0.2 0.4 0.6 0.8 1h / σ
-10
-8
-6
-4
-2
0
2
4
6
8
10
βσ2 f w
/ (2R
)
size ratio 10 20 30 50 100Derjaguin
FIG. 8: (color online) Depletion force between wall and solute for solvent bulk density ρ∗s = 0.764.
Left panel: Solute–solvent size ratio 20. Simulation data are shown by crosses (×) and circles (o)
and differ by a slightly different choice of ρw in Eq. (2) (see Refs. [31, 32] for details). Results from
explicit FMT–WBII minimization are given by the full squares and results from IE–MV are shown
by asterisks. The dash–dotted line shows the insertion route result using the WBII functional
and the dashed line gives the Derjaguin approximation. The excess surface free energy of two
parallel hard walls, needed for the Derjaguin approximation, has been calculated with the WBII
functional. Right panel: depletion force for different solute–solvent size ratios. Symbols are results
from explicit FMT–WBII minimization, lines in ascending order are morphometric results for size
ratios 10, 20, 30, 50, 100 and ∞ (Derjaguin approximation).
Eq. (16). The geometric measures ∆V , ∆A and ∆C are given by
∆V =π
3(σ − h)2 (3R − (σ − h))
R→∞≈ πR(σ − h)2 + O(R0) ,
∆A = 4πR(σ − h) − π(σ − h)2
R→∞≈ 4πR(σ − h) + O(R0) , (17)
∆C = 2π(σ − h) + π√
2R(σ − h) − (σ − h)2
(
π
2+ arcsin
(
1 − σ − h
R
))
R→∞≈ π2√
2R(σ − h) + O(R0) .
The depletion force follows as fw(h) = −∂W/∂h. In the limit R → ∞ it is given by:
fw(h < σ)
2πR
R→∞≈ p(h − σ) − 2γ − κπ
2
√
1
2R(σ − h). (18)
19
The contribution from the integrated mean curvature is subleading in 1/R but it is quantita-
tively important since it induces a singularity as h → σ which is proportional to 1/(σ−h)1/2.
This contribution precisely corresponds to the line tension term associated with the circular
apex (of length 2πr0) of the annular wedge identified in the quasi–2d analysis of Ref. [26]
and introduced in the last subsection. The line tension in the quasi–2d picture was shown
to be γ2d(ρ2d,s) and in the geometric analysis it is −πκ/2. There is good agreement between
both expressions [29]. Strictly, the singularity associated with this line term is unphysical
of course. Indeed, it can be shown that the morphometric analysis becomes ambiguous very
close to h = σ. One would require that the morphometric result for the force is unchanged
if another, slightly displaced surface around the solute sphere is chosen for the geometric
analysis. In Ref. [29], such an analysis was carried out for the sphere–sphere geometry and
the independence of the force on the chosen surface was demonstrated. However, if one
chooses e.g. the molecular surface (the surface of the set of points which is never covered
by a small solvent sphere), then this surface becomes self–overlapping close to h = σ and
one could not expect to ascribe physical significance to surface tensions and mean curvature
coefficients of such overlapping surfaces. For the sphere–wall geometry a similar analysis
holds. Note that the singularity at h = σ corresponds to a vanishing circumference of the
apex (r0 = 0), however, the 2d analysis of the density distributions at this point yielded a
small but finite value for the “effective” apex radius r0 (see Fig. 6 (right panel)).
In Fig. 8 (left panel) we show FMT–WBII and IE–MV results for the depletion force
fw between a big sphere and a wall (size ratio 20, solvent bulk density ρ∗s = 0.764) and
compare them to MC results from Refs. [31, 32] as well as to the morphometric analysis.
The force according to morphometry is plotted only until self–intersection of the molecular
surface starts. The FMT data are described very well by the morphometric results; the sharp
drop in the depletion force close to h = σ appears to mimick the behavior of the singular
term ∝ ∂∆C/∂h. The MC data suffer from relatively large error bars (except for the point
h = σ but are overall consistent with both the FMT data and the morphometric analysis.
The scatter in the IE–MV data is due to the comparatively low resolution in z–direction
as compared to the DFT data (∆z = 0.05 σ vs. ∆z = 0.002 σ). As before, all considered
IE closures give similar results, but these correspond to an overall more attractive force
compared to DFT and MC. Two other approximations for the depletion force are shown
in Fig. 8 (left panel): the DFT insertion route and the Derjaguin approximation. Both
20
approximations fail to describe the depletion force near h = σ. In the insertion route to
DFT (see Ref. [20] for its derivation), one circumvents the explicit calculation of the density
profile around the fixed wall and solute by exploiting the relation:
βW (x) = limρb→0
δF ex
δρb(x)
∣
∣
∣
∣
ρs(r)=ρw(z)
− µexb (ρs) . (19)
Here, µexb (ρs) is the excess chemical potential for inserting one solute into the solvent with
density ρs. The functional derivative of the mixture functional has to be evaluated in the
dilute limit for the solutes, ρb → 0, and with the solvent density profile given by the profile
at a hard wall (ρw(z)). This method relies on an accurate representation of mixture effects
for large size ratios in the free energy functional. Therefore, the observed deviations in the
depletion force presumably follow from the insufficient representation of higher-order direct
correlation functions of strongly asymmetric mixtures in the present forms of FMT [57]. The
Derjaguin approximation, on the other hand, corresponds to the leading term for R → ∞in Eq. (18) inside the depletion region (h < σ), but is easily extended to h ≥ σ [25, 26]:
fDerjaguinw
2πR=
p(h − σ) − 2γ (h < σ)
γslit(h) − 2γ (h ≥ σ), (20)
with γslit(h) denoting the excess surface energy of two parallel hard walls at distance h
forming a slit. According to Eq. (2), the Derjaguin force scales with R (“colloidal limit”)
which is an excellent approximation outside the depletion region. (Incidentally, outside the
depletion region all methods and the Derjaguin approximation agree with each other.) Inside
the depletion region, the neglected mean curvature (or line tension) term is very significant.
Explicit FMT–WBII results for the depletion force for size ratios up to 100 are shown
in Fig. 8 (right panel) and compared to the morphometric analysis (symbols vs. full lines).
Whereas for size ratio 10 some discrepancies are still visible, they have more or less vanished
for size ratio 100 (except for the point h = σ, see the previous discussion). Thus the
morphometric form for the depletion force (Eq. (18)) can be regarded as the appropriate
“colloidal limit”.
B. Solute–solute interaction
We now turn to the case of the depletion force between two big solute particles which are
immersed in the hard solvent. Considering the evidence from the sphere–wall case, we would
21
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h / σ
-20
-10
0
10βσ
f
-dW/dhsurface integralWBII-insWBII-morph
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h / σ
-6
-4
-2
0
2
βW
WBIIWBII-insWBII-morphDerjaguin (virial)
FIG. 9: (color online) Left panel: Depletion force between two solutes. Circles correspond to
the centered difference of the FMT-WBII data for the depletion potential (right panel), while
squares give the FMT–WBII depletion force according to the surface integral in Eq. (1). Right
panel: Depletion potential between two solutes. The curve for the morphometric potential has
been shifted by -1. The solvent density is ρ∗s = 0.8 and the solute–solvent size ratio is 5.
expect that for large size ratios, the morphometric picture reliably describes the depletion
force. Since the geometry of the overlap volume between two spheres is different from that
between a sphere and a wall, we find also a slightly different form for the morphometric
force in the limit R → ∞:
f(h < σ)
πR
R→∞≈ p(h − σ) − 2γ − κπ
2
√
1
R(σ − h). (21)
Thereby, one sees that the scaling relation fw = 2f , valid within the Derjaguin approxi-
mation, is violated through the appearance of the mean curvature (line tension) term. For
moderate solvent densities up to ρ∗s = 0.7 the morphometric form (21) gives a good descrip-
tion of the depletion force for size ratios α & 10. This has been reported in Ref. [29]. An
analysis of the full solvent density profiles between the two solutes reveals the same features
as described in Sec. IIIA.
It is interesting to investigate the regime where the morphometric description is expected
to fail. This will happen when there are strong correlations in the whole annular wedge,
i.e. where packing in both the r||– and the z–direction becomes important. Typically,
intermediate size ratios and higher solvent densities will induce these strong correlations.
Therefore, we have investigated the depletion force for size ratios 5 and 10 and solvent
22
densities ρ∗s = 0.8 and ρ∗
s = 0.9. In Figs. 9 and 10 we show results for the depletion force
and potential at ρ∗s = 0.8 for size ratios 5 and 10 respectively. For size ratio 5, the explicit
DFT calculations yield a depletion force which does not correspond to the morphometric
result at all. It is closer to the insertion route (see (Eq. 19)), though systematically lower.
This yields a depletion potential (Fig. 9 (right panel)) which is about 1 kBT or 25% more
attractive at contact than that of the insertion route. For size ratio 10, the explicit DFT data
for the depletion force reflect the morphometric form as h → σ, i.e. they give again evidence
for importance of the mean curvature (line tension) term in Eq. (21). However, away from
that regime, the depletion force deviates significantly from both the morphometric form
and the insertion route result such that the depletion potential at contact (Fig. 10 (right
panel)) is about 3 kBT or 50% more attractive at contact than that of the insertion route
(for FMT–WBII).
Note that at such a high solvent density (ρ∗s = 0.8) there are already significant deviations
between FMT–RF and FMT–WBII (Fig. 10 (right panel), circles and squares). Since FMT–
WBII has been designed to improve thermodynamic and morphometric consistency at higher
densities, the corresponding results can be assumed to be more trustworthy. As an additional
check of the numerics, we calculated the depletion force in two ways: (a) via the surface
integral in Eq. (1) and (b) via the centered difference of the results for the depletion potential
(see Figs. 9 and 10 (left panel), squares and circles). The latter is simply obtained as
W (h) = Ω[ρ(r); h]|ρ(r)=ρeq(r||,z;h) − Ω[ρ(r); h → ∞]|ρ(r)=ρeq(r||,z;h→∞) , (22)
where the grand potential Ω and the equilibrium density ρeq(r||, z; h) around the two solutes
at distance h are determined by the basic DFT equations (3) and (5), respectively. The
agreement between routes (a) and (b) is very good, save for some “fluctuations” in route (b)
close to the point h = σ where the density oscillations in the wedge are most pronounced.
The equality of both routes checks a sum rule check similar to the hard wall sum rule for the
problem of one wall immersed in solvent. Weighted–density DFT’s usually fulfill the latter
[58].
The differences between the insertion route and the explicit FMT data becomes more
dramatic for higher densities. In Fig. 11 we show results for the depletion potential for a
solute–solvent size ratio 5 at solvent densities 0.85 and 0.9 (left panel) and for a size ratio
10 at density 0.85 (right panel). For size ratio 5 the repulsive barrier has almost vanished
23
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h / σ
-50
-40
-30
-20
-10
0
10
20
30
40
βσ f
-dW/dhsurface integralWBII-insWBII-morph
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h / σ
-12
-10
-8
-6
-4
-2
0
2
4
6
8
βW
WBIIRFWBII-insWBII-morphDerjaguin (virial)
FIG. 10: (color online) Left panel: Depletion force between two solutes. Circles correspond to the
centered difference of the FMT-WBII data for the depletion potential (right panel), while squares
give the FMT–WBII depletion force according to the surface integral in Eq. (1). Right panel:
Depletion potential between two solutes. Squares correspond to FMT–WBII results, and circles to
FMT–RF results. The curve for the morphometric potential has been shifted by -2. The solvent
density is ρ∗s = 0.8 and the solute–solvent size ratio is 10.
for ρ∗s = 0.9, and the potential well at contact is almost twice as deep as compared with the
insertion route. For size ratio 10, the barrier remains (albeit at a different location) but the
well depth is equally enhanced by a factor 2 compared with the insertion route as for size
ratio 5.
The significant deviations of the depletion potential (according to the explicit FMT–
WBII results) from the previously known approximations (Derjaguin approximation and
insertion route) sheds some doubts on the quantitative accuracy of simulations of binary
hard sphere mixtures (with asymmetries between 5 and 10), employing an effective one–
component approach for the solutes interacting with their depletion potential. In Ref. [18]
the phase diagram of the mixture was scanned in that way using a virial expansion to
third order in the Derjaguin approximation for the depletion potential (see the dot–dashed
curves in Figs. 9 and 10 (right panel)). Especially for size ratio 5, the explicit DFT results
predict a depletion potential with much smaller barrier and a deeper well at contact. This
might affect the phase diagram in the corner where the packing fraction of the solutes
is low and the solvent packing fraction is high. In Ref. [59] gellation in hard sphere–like,
asymmetric mixtures is investigated with an integral equation approach (reference functional
24
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h / σ
-8
-6
-4
-2
0
2βW
ρs
*= 0.85, WBII
WBII-insρ
s
*= 0.90, WBII
WBII-ins
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8h / σ
-12
-10
-8
-6
-4
-2
0
2
4
6
8
βW
WBIIWBII-ins
FIG. 11: (color online) Depletion potential between two solutes, comparison between explicit FMT–
WBII data and the insertion route. Left panel: Solute–solvent size ratio 5 and solvent densities
ρ∗s = 0.85 and 0.9. Right panel: Size ratio 10 and ρ∗s = 0.85.
approximation [11]) to the depletion potential, akin to the insertion route. We expect
quantitative deviations at higher solvent densities compared to explicit DFT calculations
which again affect the corner of the phase diagram with low solute and high solvent packing
fraction.
Finally we want to mention that the explicit DFT methods introduced here, together with
the morphometric form of the depletion force in Eq. (21), can be used to improve available
analytic forms [60, 61] for the contact values of the pair correlation functions in hard–sphere
mixtures (note that the solute–solute contact value is given by gbb(2Rb) = exp[−βW (0)]).
IV. SUMMARY AND CONCLUSIONS
We have presented a detailed analysis of the relationship between depletion forces (solute–
wall and solute–solute) in a hard solvent and the associated density distributions around
the fixed wall/solute particles, using mainly density functional theory of fundamental mea-
sure type complemented with Monte–Carlo simulations and integral equation techniques.
For large solute–solvent size ratios, the depletion force is strongly linked to the quasi two–
dimensional confinement between the solutes (or the solute and the wall). We have shown
that the properties of this quasi two–dimensional confined system are reflected in the de-
pletion force by a line–tension term which in turn can be obtained quite generally through
25
morphological thermodynamics. Thus we can formulate an appropriate “colloidal” limit for
the depletion force in hard–sphere mixtures: Eq. (18) for the solute–wall case and Eq. (21)
for the solute–solute case. This new formulation improves significantly over the Derjaguin
approximation which is a frequently employed tool to estimate colloidal interactions in many
circumstances.
Although our analysis has been restricted to hard spheres only, the formulation of mor-
phological thermodynamics is very general such that it can be expected that the colloidal
limit for the depletion force holds also in mixtures with more general interparticle interac-
tions. However, more explicit tests in this direction are necessary.
For intermediate size ratios 5 and 10 and higher solvent densities ρ∗s ≥ 0.8 we found strong
deviations in the solute–solute depletion force between the explicit density functional results
on the one hand and the Derjaguin approximation or morphometry on the other hand. This
is a result of the strong solvent correlations in the annular wedge between the solutes. The
depletion well at solute contact is significantly more attractive (by almost a factor of two)
than previously estimated. This might have consequences for the phase diagram of binary
hard spheres at low solute and high solvent packing fractions. With recently developed
techniques [62, 63], this question appears to be tractable in direct simulations.
Acknowledgment: V.B., F.P. and M.O. thank the German Science Foundation for fi-
nancial support through the Collaborative Research Centre (SFB–TR6) “Colloids in Exter-
nal Fields”, project N01. T.S. thanks the German Science Foundation for financial support
through an Emmy Noether grant.
APPENDIX A: FUNDAMENTAL MEASURE FUNCTIONALS FOR HARD
SPHERES AND ASSOCIATED THERMODYNAMIC COEFFICIENTS
We consider fundamental measure functionals involving no tensor–weighted densities
which are defined by the following functional for the excess free energy of a hard sphere
mixture of solvent and solute particles, described by the density distribution ρ(r) =
ρs(r), ρb(r):
βF ex =
∫
drΦ(n[ρ(r)]) , (A1)
Φ(n[ρ(r)]) = −n0 ln(1 − n3) + ϕ1(n3)n1n2 − n1 · n2
1 − n3+ ϕ2(n3)
n32 − 3n2 n2 · n2
24π(1 − n3)2.
26
Here, Φ, is a free energy density which is a function of a set of weighted densities n(r) =
n0, n1, n2, n3,n1,n2 with four scalar and two vector densities. These are related to the
density profile ρ(r) by
nα =∑
i
∫
dr′ρi(r′) wα
i (r − r′) =∑
i
ρi ∗ wαi , (A2)
and the hereby introduced weight functions, wi(r) = w0i , w
1i , w
2i , w
3i ,w
1i ,w
2i , depend on
the hard sphere radii Ri = Rs, Rb of the solvent and solute particles as follows:
w3i = θ(Ri − |r|) , w2
i = δ(Ri − |r|) , w1i =
w2i
4πRi
, w0i =
w2i
4πR2i
,
w2i =
r
|r|δ(Ri − |r|) , w1i =
w2i
4πRi. (A3)
The excess free energy functional in Eq. (A1) is completed upon specification of the functions
ϕ1(n3) and ϕ2(n3). With the choice
ϕ1 = 1 , ϕ2 = 1 (A4)
we obtain the original Rosenfeld functional [3]. Upon setting
ϕ1 = 1 (A5)
ϕ2 = 1 − −2n3 + 3n23 − 2(1 − n3)
2 ln(1 − n3)
3n23
we obtain the White Bear functional [4, 5], consistent with the quasi–exact Carnahan–
Starling equation of state. Finally, with
ϕ1 = 1 +2n3 − n2
3 + 2(1 − n3) ln(1 − n3)
3n3(A6)
ϕ2 = 1 − 2n3 − 3n23 + 2n3
3 + 2(1 − n3)2 ln(1 − n3)
3n23
the recently introduced White Bear II functional is recovered.
Next we briefly recapitulate the determination of the thermodynamic coefficients p (pres-
sure), γ (hard wall surface tension), κ and κ (bending rigidity of the integrated mean and
Gaussian curvature, respectively) [6]. Consider the free energy of insertion Fb for a big solute
particle of radius Rb and associated radius R = Rb + Rs of the exclusion sphere around it.
According to Eq. (15), it is given by:
Fb = p4π
3R3 + γ 4πR2 + κ 4πR + κ 4π . (A7)
27
On the other hand, Fb is equivalent to the excess chemical potential of the solutes in a
mixture with solvent particles at infinite dilution (ρb → 0). This excess chemical potential
is obtained as the density derivative ∂/∂ρb of the mixture free energy (Eq. (A1)), therefore
we obtain:
βFb = limρb→0
β∂F ex
∂ρb
=∂Φ
∂n3
4π
3R3
b +∂Φ
∂n24πR2
b +∂Φ
∂n1Rb +
∂Φ
∂n0. (A8)
By equating the two expressions for Fb in eqs. (A7) and (A8), one finds the explicit expres-
sions for the thermodynamic coefficients as linear combinations of the partial derivatives
∂Φ/∂ni. For the Rosenfeld functional, these coefficients are given by:
βp = ρs1 + ηs + η2
s
(1 − ηs)3,
βγ = −3
4ρsσ
ηs(1 + ηs)
(1 − ηs)3,
βκ =3
4ρsσ
2 η2s
(1 − ηs)3, (A9)
βκ = ρsσ3
(−2 + 7ηs − 11η2s
48(1 − ηs)3− ln(1 − ηs)
24ηs
)
.
For the White Bear II functional, the thermodynamic coefficients have been given already
in Ref. [6] and are reproduced here for completeness:
βp = ρs1 + ηs + η2
s − η3s
(1 − ηs)3,
βγ = −ρsσ
(
1 + 2ηs + 8η2s − 5η3
s
6(1 − ηs)3+
ln(1 − ηs)
6ηs
)
,
βκ = ρsσ2
(
2 − 5ηs + 10η2s − 4η3
s
6(1 − ηs)3+
ln(1 − ηs)
3ηs
)
, (A10)
βκ = ρsσ3
(−4 + 11ηs − 13η2s + 4η3
s
24(1 − ηs)3− ln(1 − ηs)
6ηs
)
.
The coefficients of the White Bear II functional are remarkably consistent with respect to
an explicit minimization of the functional around solutes of varying radius and fitting the
corresponding insertion free energy to Eq. (A7) [6].
28
APPENDIX B: MINIMIZATION OF FUNDAMENTAL MEASURE FUNCTION-
ALS IN CYLINDRICAL COORDINATES
In the following we consider a one–component fluid of solvent hard spheres only (ρs(r) ≡ρ(r)). The equilibrium density profile ρeq(r) of the hard sphere fluid with chemical potential
µ = β−1 ln(ρs Λ3)+µex (corresponding to the bulk density ρs) in the presence of an arbitrary
external potential V (r) is found by minimizing the grand potential
Ω[ρ] = F id[ρ] + F ex[ρ] −∫
dr(µ − V ext(r)) , (B1)
which leads to
β−1 lnρeq(r)
ρs= −µ[ρeq(r)] + µex − V ext(r) . (B2)
The functional µ[ρ(r)] is given by
µ[ρ(r)] =δF ex[ρ]
δρ(r)(B3)
= β−1∑
α
∫
dr′∂Φ
∂nα(r′)wα(r′ − r) . (B4)
For the physical problems of this work, sphere–sphere and wall–sphere geometry, the external
potential V ext possesses rotational symmetry around the z–axis. Therefore we work in
cylindrical coordinates r = (r|| cos φ, r|| sin φ, z), in which the external potential and the
density profile depend only on r|| and z, V ≡ V (r||, z) and ρeq ≡ ρeq(r||, z). Eqs. (B2) and
(B4) can be solved with a standard Picard iteration procedure or a speed–enhanced scheme
(see below). The technical difficulty lies in the fast and efficient numerical evaluation of the
weighted densities nα = ρ ∗ wα and the convolutions ∂Φ/∂nα ∗ wα appearing in Eq. (B4).
Convolution integrals are calculated most conveniently in Fourier space where they re-
duce to simple products. Generically, the (three–dimensional, 3d) Fourier transforms of the
weighted densities involve a one–dimensional (1d) Fourier transform in the z–coordinate and
a Hankel transform of zeroth or first order in the r||–coordinate (see Sec. B 1). Both the 1d
Fourier transform and the Hankel transform can be calculated using fast Fourier techniques
(see Sec. B 2).
29
1. Weighted densities
The convolution integral, defining the weighted densities in Eq. (A2), is calculated by:
n(r) =
∫
dq
(2π)3exp(−iq · r) ρ(q||, qz) w(q) . (B5)
The 3d Fourier transform of the density profile ρ(r||, z) appearing in the above equation is
given by:
ρ(q||, qz) =
∫
dr exp(iq · r)ρ(r||, z) (B6)
=
∫ ∞
−∞
dz exp(iqzz)
∫ ∞
0
2πr||dr|| J0(q||r||) ρ(r||, z) , (B7)
= FT HT0 ρ(r||, z) . (B8)
Here, we have introduced shorthand notations FT for the Fourier transform in the z–
coordinate and HTi for the Hankel transform in the r||–coordinate involving the kernel
Ji(q||r||). A vector in real space is given by r = r||e|| + zez , whereas a vector in Fourier space
is given by q = q||e′|| + qzez, with e|| · e′
|| = cos φq. The 3d Fourier transforms w of the set
of weight functions w reduce to sine transforms due to radial symmetry and are explicitly
given by
w3(q) =4πR
q2
(
sin(qR)
qR− cos(qR)
)
, (B9)
w2(q) =4πR
qsin(qR) , (B10)
w2(q) = −iq w3(q) (B11)
(B12)
(The remaining weighted densities differ only by a multiplicative factor.) It is convenient to
introduce the following parallel and perpendicular components of the Fourier transformed
vector weights (k = 1, 2):
wk|| = ie′
|| · wk(q) , (B13)
wkz = iez · wk(q) . (B14)
Using these definitions, the scalar weighted densities are given by
nk(r||, z) = FT −1HT0−1
[
ρ(q||, qz) wk(√
q2|| + q2
z
)]
. (k = 0 . . . 3) . (B15)
30
The vector weighted densities, on the other hand, are given by two components (k = 1, 2):
nk(r) = nk,||(r||, z)e|| + nk,z(r||, z)ez ,
nk,||(r||, z) = −FT −1HT1−1
[
ρ(q||, qz) wk||(q||, qz)
]
, (B16)
nk,z(r||, z) = FT −1HT0−1
[
ρ(q||, qz) (−i)wkz (q||, qz)
]
.
We demonstrate this result for the example of the weighted density n2. Upon choosing
r = (r||, 0, z) we find
n2(r) =
∫
dq
(2π)3exp(−iq · r) ρ(q||, z) (−iq)w3(q) , (B17)
=
∫ ∞
−∞
dqz
(2π)3exp(−iqzz)
∫ ∞
0
q||dq|| ρ(q||, z)
∫ 2π
0
dφq exp(−iq||r|| cos φq)
−iw2|| cos φq
−iw2|| sin φq
−iw2z
=
∫ ∞
−∞
dqz
2πexp(−iqzz)
∫ ∞
0
q||dq||2π
ρ(q||, z)
−J1(q||r||)w2||
0
−iJ0(q||r||)w2z
, (B18)
which is equivalent to Eq. (B16) for k = 2. Here we made use of the integrals∫ 2π
0
dφ exp(−ix cos φ) = 2πJ0(x) ,
∫ 2π
0
dφ exp(−ix cos φ) sin φ = 0 ,
∫ 2π
0
dφ exp(−ix cos φ) cos φ = 2πiJ ′0(x) = −2πiJ1(x) .
Next we consider the evaluation of the convolution type integrals appearing in µ[ρ] (see
Eq. (B4)):
µ[ρ] =∑
α
∫
dr′pα(r′)wα(r′ − r) , (B19)
where pα = ∂Φ/∂nα and α runs over scalar and vector indices. For scalar indices, wk(r) =
wk(−r): we recover the standard convolution integral and thus:∫
dr′pk(r′)wk(r′ − r) = FT −1HT0
−1[
pk(q||, qz) wk(√
q2|| + q2
z
)]
(k = 0 . . . 3) . (B20)
In the case of vector indices, we observe that the free energy density Φ only depends on
vector densities through n1 · n2 and n2 · n2. Thus we find (k = 1, 2):
pk(r) = pk,||(r||, z)e|| + pk,z(r||, z)ez → (B21)
pk(q) = pk,||(q||, qz)e′|| + pk,z(q||, qz)ez (B22)
31
with
pk,||(q||, qz) = i FT HT1 pk,||(r||, z) , pk,z(q||, qz) = FT HT0 pk,z(r||, z) . (B23)
Therefore the vector part summands of µ[ρ] are evaluated by
∫
dr′pk(r′)wk(r′ − r) = FT −1HT0
−1[
−pk,||wk|| + ipk,zw
kz
]
(B24)
2. Fast Hankel transforms
Hankel transforms can be calculated by employing fast Fourier transforms on a logarith-
mic grid. Consider the Hankel transform
HTµ f(r||) = 2π
∫ ∞
0
r||dr|| Jµ(q||r||) f(r||) . (B25)
We define new variables x = ln(r||/r0) and y = ln(q||/q0) with arbitrary constants r0 and q0.
In terms of these variables
HTµ f(r||) = 2πr20
∫ ∞
−∞
dx Jµ(x + y) f(x) , (B26)
where Jµ(x) = Jµ(q0r0ex) and f(x) = e2xf(ex). Thus, the Hankel transform takes the
appearance of a cross–correlation integral and can be solved via Fourier transforms:
HTµ f(r||) = 2πr20 FT −1
[
FT Jµ(x) FT∗f(x)]
. (B27)
The Fourier transform of Jµ can be done analytically while the remaining ones are computed
numerically using Fast Fourier techniques.
In the actual implementation we followed Ref. [64] which details the proof of orthonor-
mality for discrete functions, defined on a finite interval in logarithmic space (and continued
periodically). The following remarks apply:
• The fast Hankel transform cannot be applied directly to the density profile, f(r||) ≡ρ(r||, z), since it does not go to zero for r|| → ∞. It is therefore advantageous to split
the external potential
V ext(r) = V (z) + Vs(r||, z) (B28)
32
into a (possibly) z–dependent background part V (z) and the remainder. The corre-
sponding background profile ρV fulfills
lnρV (z)
ρs
= −βµ[ρV (z)] + βµex − βV (z) . (B29)
The full equilibrium profile can then be written as ρeq(r||, z) = ρV (z)(h(r||, z)+1) with
h(r||, z) → 0 for r|| → ∞. The determining equation for h reads
ln(h(r||, z) + 1) = −βµ[ρV (z)h(r||, z) + ρV (z)] + βµ[ρV (z)] − βVs(z) . (B30)
Thus one needs to perform Hankel transforms only on functions which properly go to
zero for r|| → ∞. In the wall–sphere case, V (z) is naturally given by the wall potential,
Vs becomes the solute–solvent pair potential ubs and h ≡ hbs is the solute–solvent pair
correlation function. In that form, Eq. (B30) resembles the general closure for integral
equations (Eq. (10)). In the sphere–sphere case, V (z) = 0 with ρV = ρs.
• The assumed periodicity of f(x) leads to restrictions on the product q0r0 (low–ringing
condition in Ref. [64]). This condition cannot be fulfilled on one grid for both HT0 (·)and HT1 (·). However, this does not lead to any noticeable instabilities in the repeated
application of the fast Hankel transform.
• In order to avoid aliasing and the amplification of numerical “noise” in the low–x and
high–x tails of f(x) in the repeated application of the fast Hankel transform we worked
with cutoffs rmin/r0 = qmin/q0 = 0.01 and rmax/r0 = qmax/q0 = O(100). The fast
Hankel transform itself was calculated on an extended grid x ∈ [−N||∆x/2, N||∆x/2]
with either N|| = 2048, ∆x = 0.01 or N|| = 4096, ∆x = 0.005. Outside the “physical”
domain defined by xphys ∈ [ln(rmin/r0), ln(rmax/r0)] the function f(x) was put to zero
(a similar prescription applies to yphys).
3. Speedup of iterations
The density profile ρ(r||, z), the weighted densities n(r||, z) and the derivatives p(r||, z) =
∂Φ/∂n(r||, z) have been discretized on a two–dimensional grid spanning the plane (r||, z).
Spacing in z–direction was equidistant with grid width ∆z = 0.002 . . . 0.005σ with up
to Nz = 15, 000 points. Spacing in r||–direction was logarithmic (see above) with N|| ≈
33
1, 000 . . . 2, 000 points. Memory requirement went up to 16 GB for the largest grids. Com-
putations have been performed on nodes with 16 GB RAM and two Intel QuadCore pro-
cessors, with OpenMP parallelization of the arrays of Fourier and Hankel transforms. One
evaluation of µ[ρ] (one iteration) took up to two minutes.
The equilibrium density profile fulfilling Eq. (B2) or Eqs. (B29) and (B30) can be de-
termined by Picard iterations where, as a minimum requirement for convergence, careful
mixing of the current and previous iteration is necessary. However, for packing fractions
η > 0.3 and larger colloid–solvent size ratios α > 5 one easily needs several hundreds to
thousands of iterations until convergence. In view of the iteration times up to two minutes,
this is inacceptable. Therefore we employed the modified DIIS (direct inversion in iterative
subspace) scheme developed in Ref. [65] which essentially constructs the next iterative step
out of a certain number of previous steps. In the DIIS scheme, the mixing coefficients of
the previous steps, determining the solution of the next step, are obtained by a minimiza-
tion condition on the residual. The modification to DIIS consists in the admixture of the
extrapolated DIIS residual to the solution of the next iterative step, in order to enlarge
the dimensionality of the iterative subspace and thus to reach the true solution much more
quickly. Our practical experience with modified DIIS is very similar to the observations in
Ref. [65], this includes the necessity to combine DIIS with Picard steps carefully, in case the
DIIS steps show divergent behaviour. In summary, modified DIIS is a robust method for
our problem, and the total number of iterations is reduced to approximately 100 and less for
most parameter choices. The only noticeable exception occured for the case of the nearly
singular annular wedge, i.e. when the distance h between wall and colloidal sphere (or the
two colloidal spheres) is ≈ 1 σ and the solvent packing fraction ηs > 0.35. Here, several
hundred of intermediate Picard steps between two DIIS steps where occasionally necessary
to prepare the ground for the next DIIS step.
34
APPENDIX C: INTEGRAL EQUATIONS FOR THE SOLUTE–WALL CASE:
NUMERICAL SOLUTION
In cylindrical coordinates the inhomogeneous OZ equations (7)–(8) read
hss(z, z0, s) − css(z, z0, s) =
∫
ds′dz′ρV (z′)hss(z, z′, |s− s′|)css(z
′, z0, s′) , (C1)
hbs(z, z0, s) − cbs(z, z0, s) =
∫
ds′dz′ρV (z′)hbs(z, z′, |s− s′|)css(z
′, z0, s′) , (C2)
where the total and direct correlation functions hij(z, z0, s) and cij(z, z0, s) of two particles
depend on the distances z and z0 from the wall and on the projection of their position vectors
on the direction along the wall, s = r||−r||,0. The integral over s′ is a 2D convolution, which
is most efficiently done by means of the Fast Hankel transform of the zeroth-order (see
Appendix B2). In Fourier space, the OZ equations become
hij(z, z0, q||) − cij(z, z0, q||) =∑
k=b,s
∫
dz′ρV,k(z′)hik(z, z
′, q||)ckj(z′, z0, q||) , (C3)
where h[c](z, z0, q||) = HT0 h[c](z, z0, r||). The remaining z-integral is evaluated with a
simple trapezoidal rule on a uniformly discretized grid of Nz points, yielding the following
matrix equation:
Hij(q||) − Cij(q||) =∑
k=b,s
Hik(q||)RCkj(q||) , (C4)
where H and C are Nz ×Nz matrices generated by the corresponding correlation functions
at each q|| point and R is a diagonal matrix corresponding to ρV (z)δ(z − z′).
The Lovett–Mou–Buff–Wertheim (LMBW) equation (9) for the background density pro-
file in the presence of a hard wall takes the following form:
∂ρV (z)
∂z= ρV (z)
(
ρV (0)
∫
dr||css(z, 0, r||) +
∫
dz′dr||css(z, z′, r||)
∂ρV (z′)
∂z′
)
. (C5)
Here the 2D integration over r|| can be considered as the DC component q|| = 0 of the
Hankel transform of the direct correlation function css(z, z′, q||) and the remaining integral
in the second term is taken again with the trapezoidal rule. Note, that the contact density
at the plannar wall is known exactly from the wall theorem ρV (0) = βp, where p is the bulk
pressure [66]. Thus, a substitution of the contact density by the expression for the bulk
pressure provides the modified LMBW equation for the density profile [67]. Following the
ansatz of Plischke and Henderson [68], we used the Carnahan-Starling fit for pressure, which
35
is the first equation in (A10). This in general should improve the accuracy of the density
profiles near a plannar wall, if an approximation to the solvent direct correlation function
css(z, z′, r||) is applied.
The closure equation (10) is the central one in the elaboration of a successful theory,
since the OZ and LMBW equations described above are formally exact and need to be
complemented with a third relation between between correlation total and direct functions.
Unfortunately, in its exact form this third relation, expressed via additional bridge function
bij(r, r0) is defined only as an infinite sum of highly connected bridge diagrams and so cannot
be fully utilized. In practice, one needs to resort to different approximations, which are suited
for particular applications depending on the system state. We considered the Percus–Yevick
(PY), Rogers–Young (RY) and modified Verlet (MV) closures defined in Eqs. (11)–(13). For
RY we use a scaled one–parameter form for ξij, ξij = ξ/(Ri+Rj) with ξ = 0.160 to fulfill the
single–component thermodynamic consistency requirement. For ξ = 0 one recovers the PY
closure (Eq.(11)), while for ξ → ∞ one obtains the HNC closure. Likewise the HNC closure
is recovered if αij → ∞ in the MV closure (Eq. (13)). MV is consistent with PY up to the
fourth virial coefficient, if the αij are density independent. Here we follow the suggestion
of Henderson et. al. [52] and use their definition for the state–dependent parameters αij.
In the infinite dilution limit considered in the present paper these parameters reduce to the
one–component form
α =17
120ηsexp(−2ηs) + 0.8 − 0.45ηs .
The contact values from the MV closure proved to be in good agreement with the val-
ues calculated from Carnahan–Starling equation of state, which constitutes a considerable
improvement over the PY and HNC approximations.
In the application of these closures to the inhomogeneous system the following precautions
must be observed to obtain robust results.
• Technically we use the same iteration scheme as in Sec. B 3 for the numerical solution
of Eqs. (C4)-(C5) and (10) (though due to the memory restrictions the z–grid spacing
was only ∆z = 0.05σ with up to Nz = 400 points). The convergence drastically
depends on the initial conditions, i.e. the zeroth iteration. While solving the PY
closure it was enough to initialize both solvent–solvent and solute–solvent correlation
functions with zeros, the RY and MV solute–solvent correlations have to be initialized
36
with a good guess to the final solution (which was actually the corresponding PY
result). The LMBW equation imposes even stronger requirement on the initial profile,
which needs to be either close to the true profile or evolve very slowly from the flat
profile together with the corresponding correlation functions.
• It turned out that the order of iterations is crucial for the overall convergency: for a
given approximation to the density profile one needs to solve the closure and OZ equa-
tions prior to the next iteration on the LMBW equation. It is therefore very costly to
get a fully self–consistent solution of all three equations and normally the convergence
goal for the background density ρV (z) is much lower than for the correlation functions.
• To avoid the necessity of requiring the bulk equilibrium behavior in correlation func-
tions for distances far enough from the hard wall, we introduced the second wall at
L ∼ 20σ apart from the first one. The wide slit geometry allowed us to keep the z–
resolution at the maximal possible, yet feasible level. Finer resolution in z in general
improves numerical stability of the iteration procedure. Alternatively, one can use a
very recent method of the expansion into the orthogonal set of functions proposed by
Lado in Ref. [69].
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